Definition:Decreasing/Sequence

Definition

Let $\struct {S, \preceq}$ be a totally ordered set.

Then a sequence $\sequence {a_k}_{k \mathop \in A}$ of terms of $S$ is decreasing if and only if:

$\forall j, k \in A: j < k \implies a_k \preceq a_j$

Real Sequence

The above definition for sequences is usually applied to real number sequences:

Let $\sequence {x_n}$ be a sequence in $\R$.

Then $\sequence {x_n}$ is decreasing if and only if:

$\forall n \in \N: x_{n + 1} \le x_n$

Also known as

A decreasing sequence is also referred to as order-reversing.

Some sources use the term descending.

Some sources refer to a decreasing sequence which is not strictly decreasing as non-increasing or monotone non-increasing.

Some sources refer to a decreasing sequence as a monotonic decreasing sequence to distinguish it from a strictly decreasing sequence.

That is, such that monotonic is being used to mean a decreasing sequence in which consecutive terms may be equal.

$\mathsf{Pr} \infty \mathsf{fWiki}$ does not endorse this viewpoint.